Epidemic processes with immunization

We study a model of directed percolation (DP) with immunization, i.e., with different probabilities for the first infection and subsequent infections. The immunization effect leads to an additional non-Markovian term in the corresponding field theoretical action. We consider immunization as a small perturbation around the DP fixed point in $d<\mathrm{}6,$ where the non-Markovian term is relevant. The immunization causes the system to be driven away from the neighborhood of the DP critical point. In order to investigate the dynamical critical behavior of the model, we consider the limits of low and high first-infection rate, while the second-infection rate remains constant at the DP critical value. Scaling arguments are applied to obtain an expression for the survival probability in both limits. The corresponding exponents are written in terms of the critical exponents for ordinary DP and DP with a wall. We find that the survival probability does not obey a power-law behavior, decaying instead as a stretched exponential in the low first-infection probability limit and to a constant in the high first-infection probability limit. The theoretical predictions are confirmed by optimized numerical simulations in $1+1$ dimensions.